Finite Group DatabasebetaFinite Group Databasebeta| order | 16224 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C132 ⋊ C3×C4×C8 |
| Hall | #72259183 ⋊ C3 |
| Hall | #92788 ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #72210302 |
| commutator, hypocenter | C132 | C3×C4×C8 |
| Fitting | C4×C132 | C2×C3×C4 |
| socle | C2×C132 | C3×C42 |
| prime radical | C2×#11759815 | C2×C4 |
| cosocle | C4×C13⋊D13 | C22×C3 |
| LCS | G ▷ C132 | |
| UCS | C1 ◁ C4 | |
356 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^2,3,2^3,13^2|a^2=e,b^2=d,e^2=f,g^a=g^12,g^b=g^8,g^c=g^3,g^d=g^12,h^a=g^6h,h^b=h^8,h^c=h^3,h^d=h^12136 conjugacy classes — view the full character table.
| maximal subgroup of | #72259347, #72259442, #72259474 |
| maximal normal subgroup of | #72259347, #72259442, #72259474 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 3, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);